Solve each problem. See Example 5. Height of a Lighthouse The Biloxi lighthouse in the figure casts a shadow 28 m long at 7 A.M. At the same time, the shadow of the lighthouse keeper, who is 1.75 m tall, is 3.5 m long. How tall is the lighthouse?

Verified step by step guidance

1. Identify the right triangles formed by the lighthouse and its shadow, and by the lighthouse keeper and his shadow. Both triangles share the same angle of elevation of the sun at 7 A.M.
2. Set up the ratio of the height to the shadow length for the lighthouse keeper: \(\frac{1.75}{3.5}\), which represents the tangent of the sun's angle.
3. Let the height of the lighthouse be \(h\). Set up the ratio for the lighthouse: \(\frac{h}{28}\), which should be equal to the tangent of the sun's angle as well.
4. Write the equation equating the two ratios: \(\frac{1.75}{3.5} = \frac{h}{28}\).
5. Solve this equation for \(h\) by cross-multiplying and isolating \(h\) to find the height of the lighthouse.

Key Concepts

Similar Triangles

When two triangles have the same angles, their corresponding sides are proportional. In this problem, the lighthouse and its shadow form one triangle, and the lighthouse keeper and his shadow form a smaller, similar triangle. Using the ratio of corresponding sides helps find the unknown height.

Trigonometric Ratios and Shadows

Shadows are related to the angle of elevation of the light source (like the sun). The length of a shadow depends on the height of the object and the angle of the light. Understanding this relationship allows us to use proportions or trigonometric ratios to solve for unknown heights.

Proportional Reasoning

Proportional reasoning involves setting up ratios between known and unknown quantities. Here, the ratio of the lighthouse's height to its shadow length equals the ratio of the keeper's height to his shadow length. Solving this proportion yields the lighthouse's height.